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§ Approved for public release; distribution unlimited
OS oi 33 <, 'r&U'
AFIT/GA/EE/78-1
DUAL CONTROL ANALYSIS
OF AN
AIR TO GROUND MISSILE
THESIS
Presented to the Faculty of the School of Engineering
of the Air Force Institute of Technology
Air University
in Partial Fulfillment of the
Requirements for the Degree of
Master of Science
by
James P. Kauppila, B.S.
Captain USAF
Graduate Astronautical Engineering
March 1978
Approved for public release; distribution unlimited
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Preface
The intent of this study is to find a trajectory profile which will
minimize the terminal error of an air-to-ground missile. This thesis is
a follow-on study to the research conducted by Major Rony Dayan, I.A.F.
His work was concerned with evaluating the parameters of an advanced
missile guidance and control system using a maximum likelihood estimator.
This thesis was sponsored by the Avionics Laboratory, Wright-
Patterson Air Force Base, Ohio. I would like to give my sincere thanks
to Captain Gary Reid for his untiring assistance and help throughout
this entire project. Without his enthusiasm and intellectual insight,
this work would not have been possible.
My typists. Miss Patsy Rose and Miss Cheryl Gilliland, did an out-
standing job. I thank them for their assistance and professional support.
I would also like to thank my fiancee. Miss Sandra Sundermeyer, for
her patience and moral encouragement throughout this research.
ii
C □
Contents
Page
ii
List of Figures iv
List of Tables vi
Abstract vn
I. Introduction 1
Background and Motivation 1
Estimation of Unknown Parameters 3
Optimal Control Systems 6
Fundamental Concepts of Dual Control 9
Statement of Problem 9
Organization 14
II. Analysis of Problem 16
Overview 16
Development of P Matrix 19
Plan of Attack 29
III. Single Turn Analysis 33
Overview 33
Initial Turn 33
Terminal Guidance Phase 35
Case 1 37
IV. First-Order Gradient Optimization 50
Theory 50
Application to Problem 53
Case 1 59
Case 2 60
Case 3 60
V. Conclusions and Recommendations 74
Bibliography 76
Appendix A 77
I
I
Preface
Appendix B
90
VITA
1
117
List of Figures
Figure Page
1 Misalignment of Inertial Frames 2
2 Automatic Control System 7
3 Misalignment of Guidance Platform n
4 Description of Problem 12
5 Block Diagram of Phase I Flight 17
6 Block Diagram of Phase II Flight 18
7 Propagation of Error Ellipse 20
8 Geometrical Analysis of Lift Vector 22
9 Driving Sequence of P Matrix 30
10 Schematic of Flight Paths for Single Turn Analysis. 34
11 Schematic of Terminal Guidance Phase 35
12 Flight Paths for Case 1 39
13 ox Versus a for Case 1 40
14 Oy Versus a for Case 1 41
15 CEP Versus a for Case 1 42
16 Flight Paths for Case 2 44
17 Ox Versus a for Case 2 45
18 Versus a for Case 2 46
19 CEP Versus a for Case 2 47
20 Flow Chart for First-Order Gradient Technique ... 58
21 Control History for Case 1 63
22 CEP Versus Cycle-No for Case 1 64
23 PSI Versus Cycle-No for Case 1 65
24 Control History for Case 2 67
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Figure
25 CEP Versus Cycle-No for Case 2
26 PSI Versus Cycle-No for Case 2
27 Control History for Case 3 . .
28 CEP Versus Cycle-No for Case 3
29 PSI Versus Cycle-No for Case 3
Page
68
69
71
72
73
T
List of Tables
Table
Page
I
Results of
Case 1 Analysis
38
II
Results of
Case 2 Analysis
43
III
Comparison
of Case 1 and Case 2
at End of Phase I
48
IV
Results of
Phase II
Case 1 with Terminal
Maneuvering During
48
V
Results of
Gradient Technique -
Case 1
62
VI
Results of
Gradient Technique -
Case 2
66
VII
Results of
Gradient Technique -
Case 3
70
I
Abstract
Many errors are known to exist in Inertial Navigation Systems of
modern air-to-ground missiles. These error sources, if undetected, con-
tribute to navigation errors of position and velocity. This study
analyses one source of INS errors — the misalignment of the accelero-
meter reference frame. By maneuvering a missile, the error source
becomes more observable. Thus, a better estimate can be made of the
error source. This directly influences the estimate of position.
Hence, in order to minimize the terminal navigation error, some control
energy must be expended to identify the error source. This dual control
problem may be viewed as an optimization problem. By formulating a
performance index of the terminal error and control energy appropriate
mathematical techniques should yield an optimal flight trajectory.
This thesis seeks to analyze the dual control nature of an air-to-
ground missile. Two methods are used. The first uses a predetermined
flight path which is incremental until a minimum is reached. The second
is a first-order gradient which allows greater freedom in the control
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law.
vii
DUAL CONTROL ANALYSIS
OF AN
AIR TO GROUND MISSILE
I. Introduction
Background and Motivation
The production of low cost, expendable, air-to-ground missiles is
of primary consideration in the development of an effective standoff
weapons systems capability. Since these systems are intended strictly
for one time use only, low cost navigation systems are employed. These
navigation systems are subject to a variety of error sources such as
bias errors, scale factor errors, initial position and velocity errors,
gravity anomalies, and transfer alignment errors. Each of these sources
contribute to errors in the final position of the missile. If the
terminal errors are large enough the missile will miss the target com-
pletely. It is therefore advantageous to find methods to reduce all
sources of error to a minimum.
The transfer alignment problem is perhaps the single most signifi-
cant source of error for low cost navigation systems. Even though the
carrier aircraft is flying in straight and level unaccelerated flight
there are inherent vibrations, wind gusts, and turbulence that create
problems in aligning the platform of the inertial guidance system. If
this misalignment is too large, then there will be incorrect values of
specific force measured by the accelerometers. This will be used in
the navigation computer and thus erroneous values of position and
velocity will result. Figure 1 shows schematically the problem of mis-
alignment of the inertial reference frames.
1
Figure 1. Misalignment of Inertial Frames
If an effective estimation algorithm is used to accurately predict
the value of the misalignment angles then this information can be relayed
to the navigation computer to arrive at a much improved estimate of
position and velocity. This in turn would greatly enhance the proba-
bility of a successful "hit" of the target.
In a previous study, conducted by Major Rony Dayan I.A.F., the
problem of estimating the misalignment angles was undertaken. By
choosing an appropriate system model and using radar tracking the mis-
alignment parameters can be estimated. However, it was not determined
if the estimation can be improved by maneuvering the missile. This is
the central theme of this study: that by maneuvering a missile it may
be possible to induce large sensitivities and hence improve our esti-
mation capability.
In order to establish a clear understanding of the problem some
basic background material concerning estimation, optimal control systems,
and dual control will be presented.
2
Estimation of Unknown Parameters
In order to gain understanding and explain the processes of natural
and man-made environments, models are created. Models in this sense are
mathematical descriptions oj. ,ese processes. They may pertain to any
system. The system may be physical or nonphysical. The important point
is that models are made to explain the dynamic processes of the system.
Mathematical models aid in understanding the performance of the system.
By selecting a suitable criteria, it is possible to select an input which
will optimize this factor. By optimal it is meant that the performance
criteria will be maximized or minimized.
Modeling encompasses four problem areas: representation, measure-
ment, estimation, and validation. Representation deals with the mathe-
matical structure of the system. Is it static or dynamic, li.: ar or
nonlinear, discrete or continuous, deterministic or stochastic? Measure-
ment deals with the physical quantities of the system. There are two
basic types of physical quantities: signals and parameters. It is
difficult to give a precise definition of signals and parameters because
many times they tend to overlap one another. Basically signals are time
varying quantities which can easily be measured and parameters are con-
stants which are known only to a certain degree of accuracy. Take for
example the relation
F(t) = Ma (t) (1)
If we apply a known force and measure the acceleration then force and
acceleration are the "signals" and the mass is the unknown "parameter."
With accurate measurements of the signals, accurate estimations may be
made of the parameters. However, in most systems there is a certain
amount of "noise" present. This presents a degree of uncertainty in
3
our measurements. In this case, the uncertainty is described by the
covariance of the measurement. This would be the stochastic case. It
is desirable to reduce the amount of uncertainty to a desired level.
Exact measurements are simply not possible. In summary, parameter
estimation is the determination of those physical quantities that cannot
be measured directly but can be determined from quantities that can be
measured.
Parameter estimation, sometimes referred to as parameter identifi-
cation, encompasses a large block of engineering. Depending on the type
and structure of the system, it may or may not be possible to identify
the parameters.
Many numerical techniques for parameter identification are based
on parameter sensitivity. Parameter sensitivity is the study of any
property of a mathematical model which might be altered by a change of
the parameter values from their nominal or assumed values (8:1). If a
system can be described by a nonlinear differential equation of the form:
X(t,b) = f (x,b,t) (2)
where b is the unknown, constant parameter, then the relationship may be
linearized to the form:
dt $ (t’bo) = If (x*b’t) i bo * (t,bo) + If (x’b>t) i bo (3)
where
*<t,bo) =^b (t,b)|bQ (4)
Equation 3 is known as the "sensitivity system" and equation 4 is
known as the "sensitivity function" (8:4). Parameter sensitivity may
be defined as the Frechet derivative of the unknown mathematical model
output with respect to the unknown constant parameter (8:1).
4
One approach to parameter identification is to view it as an opti-
mization problem in which parameter values are selected to either maximize
or minimize some selected cost function. This may be the difference
between some predetermined model output and the actual measured output.
Generally an iterative technique is used and involves the use of parameter
sensitivities. If our sensitivities are "large" then there will be a
noticeable change in our output and we can compute our parameter values
more accurately.
It is apparent from the discussion that parameter identification
is extremely important if it is desired to obtain an accurate mathemat-
ical model. In the case of navigation, parameter values of the equations
of motion are the unknown sources of error. Such things as bias values,
scale factor errors, and misalignment angles are known to exist in any
Inertial Navigation System (INS) to some degree. If these values are
estimated incorrectly then there will be errors in the model output of
position and velocity. Thus for an accurate INS, it is necessary to be
able to estimate parameters accurately.
In the navigation problem discussed above, accurate state estimation
is also required. As described by Eykoff (3:446), a problem of this type
with constant parameter vector a may be reformulated as:
x _ f (x,u,a,u,t)
L*J L 0 J
(5)
Thus, even for a process with linear dynamics and linear in the
parameter, the combined parameter and estimation problem is nonlinear.
This suggests the use of an iterative technique for the solution of this
problem. One of the methods suggested by Eykoff is a quasi-linearization
approach.
5
In many cases the problem of optimizing performance by controlling
the input variables is considered. When the system contains unknown
; parameters, the problem becomes a tradeoff between parameter estimation
and optimal control. This is the theory of dual control presented by
Feldbaum in 1960 (4:31). Before discussing this, however, a brief review
of optimal control systems will be presented.
Optimal Control Systems
In Figure 2, a block diagram of a general automatic control system
is presented (4:8). "B" represents the controlled object. "A" is the
controller. The controller provides the input, U, to the controlled
object B. "X" represents the controlled variable which characterizes
the state of the controlled object B. The perturbations, Z, are mea-
sured as input noise and cause the output X to vary from the desired
state.
The controlled object is fixed for a particular system but the
controller may be selected from a large class of possible algorithms.
When considering optimal control systems, the problem is to choose the
controller, A, which will control B, in a known, predictable manner to
minimize the performance criteria. Take for example the control of an
automobile. The performance function may be efficiency or miles per
gallon. In this case it is desired to maximize the function. The
driver would be the controller and using his knowledge of speed and
efficiency, traffic conditions, route select 'n, and maintenance required,
he will be able to control the car at the optimal level.
In selecting a controller usually the following factors are used:
1. Characteristics of the object B
6
= vector of initial input conditions
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f
Figure 2. Automatic Control System
2. Demands on object B
3. Characteristics of the information about B entering A
The characteristics of the object B are the relations between the
input and output. In general operator notation this may be written as
X - f(u,z) (6)
or for dynamic systems
X - f(x,u,z,t) (7)
In any realistic system there will be some constraints. Usually the
control u may be restricted such as:
lu1l<u1 . .
• lu |<u
m m
where U^. . . U^ are selected constants. The state of the system may
also have constraints, especially at the terminal state. These may be
7
written as
*P [x.tf] = 0 (9)
or more specifically in the form
xL(tf) - xlf = 0
x2(tf) - x2f = 0 (10)
• • #
xn#(tf) ~ Xnf = 0
The demands on the controlled object B are characterized by the
selection of the optimality criteria. Usually a minimum or maximum is
desired. Typically J is selected based on terminal conditions and an
integral term of the form:
J = 0 (x,tf) + L(x,u,t) dx (11)
The actual selection of the optimality criteria presents a difficult
problem in itself. The important point is that once J has been selected
the problem is finding the input control u to minimize the function.
The characteristics of information entering A vary significantly
among systems. Some systems may have complete information about the
controlled object, while other systems may have only partial information.
It is the latter that presents the most concern. With only partial in-
formation about the controlled object B some of the control action must
be used for learning more about the nature of the object itself and not
just strictly minimizing the performance function. This is the under-
lying assumption of dual control discussed in the following section.
8
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Fundamental Concepts of Dual Control
When a system with incomplete knowledge of the controlled object B
exists, the controller A is attempting to solve two problems (4:26):
1. To learn more about the characteristics of the controlled
object B.
2. To determine future control actions to minimize the performance
criteria.
Feldbaum (4:27) offers an analogy of a man interacting with his
environment. Man studies the surroundings in order to influence them
in a direction useful to himself. However, in order to direct his own
actions better he must have a better understanding of his environment.
Therefore, sometimes he acts on the environment not to take advantage of it
but only to try to understand it better.
In considering a system with unknown parameters, the conflicting
goals in the control law are to learn about the parameters and to direct
the object in a manner to minimize the performance function. As a result
the control law must have characteristics of distributing its energy for
learning and achieving the performance objective (1:2).
As a measure of the information content of the object B, a probability
distribution of the characteristics may be used. A reduction in the
covariance of the parameter estimates is a measure of the amount of
learning made by the dual control algorithm. Thus, in the case of an
air-to-ground missile, the control action should be able to demonstrate
the relationship between knowledge of the error parameters and the esti-
mates of the navigation states.
Statement of Problem
This study will analyze an inertially guided air-to-ground missile.
9
with an unknown but constant misalignment angle of the guidance platform.
Figure 3 shows a simplified drawing of the guidance platform. The overall
objective of the analysis is to find the nominal trajectory which will
minimize the terminal position covariance as indicated by the circular
error probable (CEP).
Figure 4 shows the overall scenario of the problem to be considered.
There are two distinct phases of flight. Phase I is a period in which
the carrier aircraft can track the missile by radar. There is also a
data link between the missile and aircraft which allows for updating
state and parameter estimates. Phase II is without radar tracking or
communications data link. The missile is controlled by pure inertial
guidance. During this phase parameter estimates of bias values, scale
factor errors, and misalignment angles remain at the last estimate made
during Phase I.
The terminal target is fixed in space but the missile is free to
maneuver during flight. This is especially true during Phase I where
parameter estimates occur.
The objective of the study, as stated earlier, is to find the
nominal trajectory profile which gives, a priori, the lowest terminal
position covariance. This is measured by the CEP which can be written
as a function of the position covariance as:
CEP - 0.588 (ox + ay) (12)
However, a secondary objective is to limit the amount of control used so
the total performance objective is of the form:
J - CEP +/'tf WjL2 dt (13)
*'to
where w^ is a constant weighting value determined from engineering
judgment. The major emphasis on J will be on the CEP; however, a
10
1
Figure 4. Description of Problem
realistic problem must also be concerned with a finite energy source
for control input.
This problem has several characteristics which deserve explanation.
During Phase I the maneuvering of the missile allows estimation of the
system model parameters. The model equations for acceleration, to be
developed in detail in Chapter II, are:
1 -p ax
V 1 ay
(14)
ax, ay “ specific force measurements
It is apparent that by commanding inputs to the x and y accelerometers
12
the misalignment angle, p, becomes more observable. Hence, a better
estimate of p may be made by maneuvering the missile in some matter.
Phase II of the flight is "open loop"; that is, no more parameter
estimates are fed back to the missile INS. Hence, any uncertainty which
may still exist during Phase II will directly contribute to the success
or failure of the missile system.
Considering these characteristics, it may be concluded that this
problem is one of dual control. This is because during Phase I use of
the control input directly contributes to the knowledge of the system
parameters and hence influences the ultimate objective — to get to the
target with the minimum CEP.
Approach to the Problem
This study could logically include all phases of flight from boost
to cruise and terminal guidance to the target. However, since this
study is primarily a feasibility study in applying the concept of dual
control to an air-to-ground missile, the approach will be to use a
simplified analysis. Several assumptions will be made which will limit
the modeling process but still allow a detailed study of the dual control
concept. These assumptions are:
1. The missile has boosted to cruise velocity and maintains a
constant velocity of 1000 m/sec.
2. The missile is restricted to maneuvering in a horizontal, two-
dimensional plane. Hence, gravity accelerations are not considered.
3. The control of the missile is restricted to deflection of the
control surfaces and the resultant lift vector is always perpendicular
2
to the velocity vector. The lift vector is restricted to 100 m/sec
maximum.
13
4. The accelerometers are misaligned by a small angle y which is
assumed constant. See Figure 3 for a description of the missile platform.
5. During Phase I of flight, the missile is tracked by radar from
the carrier aircraft. The aircraft is modeled as a stationary point mass.
6. During the tracking phase, continuous measurements are taken of
missile position. The radar measurements are taken directly in polar form
but can be related to cartesian form by a simple transformation.
These assumptions simplify the analysis; however, if the concept of
dual control can be successfully demonstrated then the door will be open
to future study of this concept on a more detailed basis.
Two algorithms were employed to solve this problem. The first, a
"single turn" analysis, commands the missile to turn to a desired a angle
then fly to a specified range limit. It then flies to the target by
turning to the proper heading. This technique, although simple in con-
cept, allows for the numerical demonstration of the theory.
The second algorithm is a modified gradient technique. By searching
in the negative gradient direction, the method should converge to a local
minimum. This technique allows for greater flexibility in the amount of
maneuvering during Phase I.
Organization
This thesis is organized into five chapters. Chapter I is the intro-
duction, motivation, and underlying background material concerned with
the problem. Chapter II formulates the problem in detail and describes
the plan of attack. Chapter III discusses the single turn analysis for
solution. This chapter brings together the theory of Chapter I and prob-
lem of Chapter II into a realistic numerical example. Chapter IV dis-
cusses a more sophisticated numerical technique for finding a solution.
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Gradient optimization was used because of its fast rate of convergence
during the initial phases of solution. The numerical difficulties of
this method are also explained in this chapter. The last chapter sum-
marizes the results, forms conclusions, and makes recommendations for
further study. There are two appendices. Appendix A presents the
details of the continuous measurement covariance equation of the Kalman
filter. Appendix B presents a listing of the computer programs used.
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II. Problem Formulation
Overview
The problem to be undertaken by this study is to find a control
input which will minimize the terminal position error of an air-to-
ground missile employing a low cost inertial navigation system. There
are two inherent difficulties which preclude an easy solution. These
difficulties are the time-of -flight and the knowledge of the misalign-
ment angle. During Phase II of the flight, the covariance of the
position estimates can only increase with time. Therefore, the time-of-
flight must be kept to a minimum. Knowledge of the misalignment angle,
y, helps directly in reducing the covariance of the position estimates.
Knowledge of the misalignment angle is achieved by maneuvering the
missile. However, too much maneuvering will increase the time of flight.
Therefore, the optimum solution will be a tradeoff between the amount of
maneuvering versus the time of flight. In summary, the main objective
of this study is to find a control law which will minimize the CEP of an
air-to-ground missile with a small, constant misalignment angle of the
INS platform. The control law will have to consider the conflicting
difficulties of:
1. Time of flight: the shorter the time of flight the smaller the CEP.
2. Maneuvering: increased maneuver ing during Phase I directly aids
in reducing the CEP.
As described in Chapter I, the missile trajectory will be divided
into two phases of flight. During Phase I radar measurements will be
taken. The estimates of position and velocity will be a result of
measurements from the INS accelerometers and the radar measurements
themselves. Figure 5 shows a block diagram of Phase I flight. The
16
Figure 5. Block Diagram of Phase I
important feature of Phase I is that the state estimates and covariance
matrix are a result of both the INS specific force measurements and the
radar tracking measurements. As a result, knowledge of the misalignment
angle y is constantly being updated and improved.
Figure 6 shows a block diagram of Phase II flight. During Phase II,
no radar measurements are taken. The state estimates are produced
strictly by the specific force measurements taken by the INS platform.
The important feature during Phase II is that the misalignment angle y
is not being updated but remains at the last estimate, y , made during
Phase I.
17
Our main objective is to minimize the CEP. This can be achieved by
minimizing the terminal position covariance of the estimate. Covariance
is defined as (6:80):
cov(x,y) = Z (Xi - yx) (Yj - Uy)h(Xi,Yj)
= E [(X - yx) (Y - yy)] ( 15 )
= E [XY] - yxyy
where
yx, yy = mean value of x,y
h (Xi, Yj) = joint probability function
Covariance is a measure of the uncertainty of the random quantity involved.
For a Graussian distributed random variable, 67.8 percent of the random
18
samplings of that variable will be contained within ±1 standard deviation
(0) of the mean value. Obviously the smaller the o the smaller the dis-
>
t
persion about the mean. It is therefore desirable to minimize the covar-
iance for a successful missile.
The covariance of the x and y estimates may be thought of as an
error ellipse. As shown in Figure 7, the initial covariance of x and y
is an area of uncertainty of the estimate. The ellipse is propagated
forward by a Kalman filter during Phase I. This will be developed
mathematically in the next section. Now it is sufficient to say that
the covariances of x and y are a function of the lift and time of flight,
and misalignment angle y:
cov(x) = f(L,tf,yU)
cov(y) = f (L, tf ,jU)
(16)
During Phase II, the covariance propagation is expressed as a linear
system driven by white noise.
The two flight paths shown in Figure 7 show the intuitive effects
on the error ellipse. In flight path A no lift is produced hence the
misalignment angle y is not observable. In flight path B the lift
generated directly effects the observability of y and hence aids in
reducing the covariance.
L — ►u — ►cov(x), cov(y)
The covariance is represented by the P matrix. This matrix is symmetric
and positive semi-definite. The derivation of this matrix is presented
in the following section.
Development of P Matrix
The dynamical relations of the missile position may be expressed as:
19
Figure 7. Propagation of Error Ellipse
Xx = V1 = V Cos 0 (17a)
x2 = v2 = v Sin 0 (17b)
Assuming 0 is the angle between the velocity vector V and the axis.
Since the lift vector is assumed to always be perpendicular to the
velocity vector, the change in heading angle, 0, may be expressed as:
0 = L/V (18)
It was also assumed that the missile platform is misaligned by a
small angle, y. This means incorrect specific force measurements will
be made by the accelerometers and this incorrect information will be
relayed to the on-board missile navigation computer. More specifically
looking at the lift vector, L as shown in Figure 8, L^, is the desired
*
component of lift in the X^ direction and is the actual component
*
of lift measured by the X^ accelerometer. As long as y is small
20
4
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and However, the discrepancy is significant enough to cause
the missile to navigate incorrectly. The equations for acceleration in
the and X2 directions are respectfully
Sx = V = -V Sin 0 0 = -V Sin 0 * L/V (19a)
= L Sin 0
Sy = V2 = V Cos 0 0 = V Cos 0 ‘ L/V (19b)
= L Cos 0
These equations represent the desired or "truth model" representation
of X. Because of the misalignment problem, V may be expressed differently.
The relationship between the measurements in a non-raisaligned frame and
one which is misaligned may be considered as a coordinate transformation.
Figure 8 shows the geometrical interpretation of the lift vector. In the
nominal frame, the lift vector is:
L *
(20)
In the perturbed frame, the lift vector is:
LL2*
(21)
These two frames are related by the direction cosine matrix:
— sin ]
cos y — sin y"l
sin y cosyj
L*
(22)
thus
*= Cos y - Sin y (23a)
L2 ■= Sin y Lj* + Cos y L2* (23b)
For small angle assumption, and this results:
Sx ■■ - y L2 + £ x (24a)
21
«
1 1
Figure 8. Geometrical Analysis of Lift Vector
sy
y L1 + L2 + ^ y
where £x, £y are the noise inherent to the accelerometers,
in matrix form:
N » T1 - yl P
J + N
N Ly L1
- 2. J N
or in first order form:
(24b)
(25)
22
0 0 10 0
0 0 0 1 0
0 0 0 0 -L,
t
0 0 0 0 Lj
0 0 0 0 0
t—;
i
0
0
CM
X
0
0
V1
+
L1
+
«1
<N|
>
L2
^2
L-.
0
0
(26)
Equation 26 represents the dynamical relations of a missile at a constant
velocity with a constant misalignment angle, y. The last term of the
equation represents noise or undesired disturbances present to some
degree in all systems. The equation is of the general form:
x*=Ax + Bu+G£
The covariance of the measurements of position, velocity, and of
the misalignment angle y can be expressed by the covariance matrix, P.
In this case P is a 5 x 5 symmetric matrix of the general form:
(27)
During Phase I, the P matrix is propagated by a Kalman filter using
INS data and radar measurements. The equation describing P during
Phase I is (7:273):
P(t) » F(t)P (t) + P(t)FC(t) - P(t) H R_1 H P(t) + G Q GC (28)
where
[P]
11
P12
P13
P14
P15
21
P22
P23
P24
P25
31
P32
P33
P34
P35
41
P42
P43
P44
P45
51
P52
P53
P54
P55
F(t) • "A" matrix of system equations
H = output position matrix
23
R = measurement covariance matrix
I
.
)
G = noise position matrix
Q = noise covariance matrix
The F matrix has already been developed as the "A" matrix of the
missile INS system.
The radar measurements are taken directly in polar form, that is,
range and angle information. Inherent to any radar system is a covariance
associated with range and angle measurements. For this problem, these
values are chosen as:
CfR = 100 m (29a)
oG = 10 ^ radians (29b)
However, since it is desirable to work in cartesian coordinates, expres-
sions must be developed to relate the R matrix to the aR and oQ values.
The R matrix can be written as:
ax2
axy
axy
ay2
R =
The x and y positions are related to the radar measurements by:
x = r Cos 0
y = r Sin G
2
Thus, an expression for Ox^ may be developed as follows:
(30)
Ola)
(31b)
To first order, the Taylor series approximation of Ax is:
Ox,
0X.
Ax - — 1
ar
Ar
+ A6
(32)
« Cos G
Ar
- r Sin GAG
(33)
Ox2 - E[Ax1'Ax1] =E[(Cos 0 R - R Sin GAG) (Cos GAR - R Sin GAG)] (34)
24
= E[Cos2OAR2 + R2 Sin20A02 - 2R Sin 0 Cos 0A0 R]
= Cos20 E[AR2] + R2 Sin~20 e[A02] - 2R Sin 0 Cos 0 E[A0AR]
For uncorrelated and zero mean functions e[A0Ar] = 0, therefore.
1 2 2 2 2 ^
0X2 = Cos 0OR + R Sin QOq
(35)
Similarly for ax^:
oX, OX
AX2 = ^ AR + 50 AG
(36)
AX2 = Sin 0 AR + R Cos 0A0
(37)
ox, = E[AX2*AX2] = E[(Sin0AR + R Cos0A0) (Sin 0AR + R Cos 0A0)] (38)
E[Sin20AR2 + R2 Cos20A02 + 2R Sin 0 Cos 0ARA0]
= Sinz 0 E[ARZ] + Rz CosZ 0 e[Cos20A02] + 2R Sin 0 Cos 0E
AR
A0
2 2 2 2 2 2
Ox, * Sin 0 oz + R Cos 0 o^
Z R u
(39)
Similarly for ox^x2:
aX^ *= E[AX1-AX2]
(40)
0x^2 = E[Cos 0 Sin 0AR2 - R2 Sin2 0A0AR + R Cos2 0ARA0
(40a)
Sin 0 Cos 0A02]
0x^2 *= Cos 0 Sin 0 E[AR2] - R2 Sin 0 Cos 0 E[A02] (40b)
2 2 2
OXjX2 » Cos 0 Sin 0 oR - R Sin 0 Cos 0 o^
(40c)
Now an expression for R ^ may be developed
„-l adl R
R “TaT
(41)
25
,-l
ox.
-0x^X2
-0x^2 Ox^
2 2 2
(ox^ Ox2 - ox.^ x2)
(41a)
letting
2 2 2 2
A = ox 2 / (ox^ 0X2 - ox^ x2)
2 2 2
B = 0x^2 / (Ox^ Ox 2 - Ox^ x2>
2 2 2 2
D = Ox^ / (Ox^ OX2 - OXj^)
(41b)
(41c)
(41d)
the matrix may be written as:
-■[: :]
(42)
Again noting that the P matrix is symmetrical only the upper diagonal
elements are necessary to propagate. After multiplying the matrices
developed in Equation 28, the following equations are derived for
Phase I:
'11 ■ 2pn ' pu <AP11 + BP12> - P12 <BP11 + dp12>
P12 ° p23 + P14 “ P11 (AP12 + BP22) " P12 (BP12 + DP22)
P13 “ P33 ” L2P15 ~ P11 (AP13 + BP23) ~ P12 (BP13 + DP23)
P14 = P34 + L1P15 ' P11 (AP14 + BP24) " P12 (BP14 + DP24>
P15 " P35 ' P11 (AP15 + BP25) " P12 (BP15 + °P25)
P22 ” 2P24 " P12 (AP12 + BP22^ ~ P22 (BP12 + DP22)
(43)
P23 “ P34 ' L2P25 “ P12 (AP13 + BP23* “ P22 (BP13 + DP23)
P24 “ P44 + L1P25 " P12 (AP14 + BP24) ' P22 (BP14 + DP24>
26
P25 " P45 " P12 (AP15 + BP25) " P22 <BP15 + DP25>
P33 = "2L2P35 “ P13 (AP13 + BP23) “ P23 (BP13 + DP23)
P34 " "L2P45 + L1P35 " P13 (AP14 + BP24> " P23 <BP14 + DP24)
P35 = -L2P55 "P13 (AP15 + BP25) ' P23 (BP15 + DP25)
P44 " +2L1P45 " P14 <^14 + BP24> " P24 (BP14 + DP24)
P45 = +L1P55 " P14 (AP15 + BP25) " P24 (BP15 + DP25)
P55 = ~P15 (AP15 +BP25) " P25 (BP15 + DP25)
During Phc.se II, no radar measurements are taken so the equation for P
is:
P(t) = F (t)P(t) + P(t)Ft(t) + GQGt
Thus during Phase II, the P equations become:
(44)
P = 2P
11 13
P — P 4- P
12 *23 *14
P = P —TP
13 33 L2 15
P14 = P34 + L1P15
P *= P
15 35
P = 2P
22 ^ 24
P * P —TP
23 34 V25
P24 “ P44 + L1P25
P » p
25 45
(45)
P33 “ ~2L2P35
27
The main objective is to minimize the covariance of x and y, or
the and P 22 states, at the terminal time. By looking carefully at
the equations the interaction between the covariance of the error, P,.,.,
and the P^ and P22 states become evident. The driving sequence is
shown in Figure 9. It is therefore apparent that knowledge about the
error parameter will directly effect the P ^ and P22 states. It is
also shown that lift is necessary to influence P^ and P 22' t*ie
nominal, "no-lift" case, both and L2 are 0 and P,.,. will not effect
P^ or P22. Therefore, it is only through maneuvering the missile that
P^^ will effect P^ and P22>
The system state equation may be augmented by the P equations to
the form:
(46)
By assumption 1, the magnitude of velocity is constant, only the
heading angle, 9, needs to be propagated. Therefore, the final system
of Interest is as follows:
28
V Cos 0
j.
X =
— —
X1
X
•
•
X2
Y
•
•
X3
0
X4
=
pn
=
X5
P12
•
•
>
_P55_
V Sin 0
L/V
2Pn - P11<AP11 + BP12) - P12<BP11 + DP12> <47>
This 18 state vector equation is the basis for the numerical techniques
to follow. By choosing appropriate initial values and a proper method
for selecting the control vector L then the objective of minimizing P
and P22 will be achieved.
The state equations are nonlinear so an appropriate numerical
integration method will be used.
Plan of Attack
The first step is to formulate a performance objective, J. Generally,
the cost is of the form:
where
/-tf
J * 0[x,t^] + / L(u,x,t)dt
*/to
0 * terminal condition of state
vector at t^
L * performance criteria of control
input
(48)
29
Since the main objective is minimization of the CEP, the cost function
will be
where
J = 0.588 (Ox + Oy) | + f W.L^dt
ax = covariance of x estimate
= (P )**
v 11'
ay = covariance of y estimate
= (P )**
22)
(49)
.-5
= selected arbitrary weighting value = 10
The integral term for this problem is just a quadratic function of
the control input only
L(x,u,t) = WXL2 (50)
in this case the dimension of m, of the control vector u is 1. As
described in Chapter I, |u^|<u^ for any practical system. Choosing a
maximum of a lOg turn for the missile
L = 100 m/sec2 (51)
max
Two methods will be used to determine the minimum of the cost
function. The first will be a single turn analysis. In this concept,
the missile is commanded to turn to a predetermined heading angle, a,
fly to the radar limit, then turn to the target. Then the performance
function is calculated. By increasing a slightly the technique should
converge to an optimum a angle. This method does severely limit the
amount of maneuvering but should demonstrate the dual control concept.
The second method will be a numerical gradient technique. This
31
will allow for greater maneuverability in searching for a minimum of the
cost function.
32
III. Single Turn Analysis
Overview
The basic approach of this method is to find an optimal flight path
based on an initial heading angle, a. The system equations, as described
in Chapter II, are:
£ = f(x,u,t)
(51)
and the performance function J is:
CEP +
f
f 2
W L dt
(53)
Figure 10 shows the schematic idea behind the single turn concept.
Beginning with an initial turn rate, the missile is commanded to turn
to a desired a angle. It flies a specified distance then begins a turn
to intercept the terminal condition. Thus, the maneuvering will be
accomplished during Phase I of the flight. The cost function is then
evaluated. The process is repeated after incrementing a. The pro-
cedure is continued from a = 0 to a = 1.0 radians to show the general
trend of the CEP versus the amount of maneuvering.
As seen from Figure 10, the amount of lift and the time-of-f light
required will vary greatly between flight path A and flight path D.
Therefore, the optimum flight path will be a tradeoff between these two
factors.
Initial Turn
To provide a valid comparison between no-maneuvering and maneuvering
an initial a angle of 0.0 was chosen. The increment size is 0.1 radian
2
to a maximum of 1.1 radians. The maximum permissible lift is 100 m/sec
which yields a maximum turn rate of 0.100 rad/sec. To insure the proper
33
Radar Range Limit
Figure 10. Schematic of Flight Paths for Single Turn Analysis
2
amount of turn, the commanded lift remains at 100 m/sec until (a - Q)
<0.100. The following scheme was used to command the initial turn:
L *= 1000 (a - 0) (54)
where
with restriction
0 ■ Current Heading Angle
Thus, if o =0.5 and 0 = 0.45 then L = 50 m/sec^. By using an inte-
gration step size of At * 1.0 sec, this scheme will insure an accurate
34
■V.
turn. After the completion of the turn, no lift is commanded so the
missile flies in a straight line to a predetermined range of 25,000 ra.
prior to turning towards the target.
Terminal Guidance Phase
It is during this phase of flight that the lift vector is commanded
in such a manner to insure the terminal condition is met. Figure 11
shows the geometrical consideration for the guidance law. The scheme
is centered on the 9r angle. If the current 9 angle can be made equal
to the angle then the missile will be heading directly toward the
target. Mathematically, the objective is to drive (9 - 9p ) to zero.
From Figure 11, two equations may be formulated based on position
at the 25,000 m. range and estimating the final time tf.
These equations
are:
V Cos 0
req
(*f " X-^t))
(£f - t)
(55)
V Sin 0 = X2(t)
req
(£f-t)
(56)
By squaring each side, an expression for £f may be found.
22 2 (xf ~ x. (t))2 + x 2(t)
V (Cos 0 + Sin 0 ) = — ±
req req (tf - t)2
(57)
V2 = Xf2 - 2XfX1 + Xx2 + X22/(tf - t)2
£f2 - 2t£f + |t2 - [Xf2 - 2XfX1 + X,2 + x22]/v2J = 0
(58)
(59)
using the quadratic formula and taking the positive root tf becomes:
£f = 2t + ^4t2 - 4t2 = 4[Xf 2 - 2XfXx + X12 + X22]/V2
(60)
letting
RAD = ^4(Xf2 - 2XfXx + Xj2 + X22)/V2
(61)
£f = t + RAD/2
(62)
knowing the current the 0 is:
f req
0req “ Cos_1((Xf - X1)/((tf - t)V))
(63)
The lift required, , can now be calculated from
0 - 0
L --5S3 V
req At
With constraint
(64)
36
|L If 100 m/sec^
1 max 1
This algorithm will turn the missile at the maximum rate to inter-
cept the target in minimum time of flight. To maximize the maneuvering
during Phase I, this turn is initiated at a range of 25,000 m. , thus the
turn will be completed in Phase I and during Phase II only a minimum of
lift will be required to maintain a target interception course.
To investigate the problem using this method, two cases were chosen.
Case 1 has a radar range of 50,000 m. and terminal condition of X^ =
101,000 m. The initial range is 1000 m. thus for a = 0 the time of
flight is 100. sec. For Case 2, the radar range is decreased to 40,000 m.
and the terminal condition is = 201,000. The initial range remains
at 1000 m. thus for a = 0 the time of flight is 200. sec. The results
of these two cases are presented in the following section.
Case 1
Radar Range = 50,000 m.
Xf = 101,000. m.
X* = 1,000. m.
The major results for Case 1 are presented in Table 1. With no
maneuvering, a = 0, the terminal CEP is 14.00. This decreases to 8.05
at a =0.1 and is the minimum value for Case 1. This shows a definite
relationship between maneuvering, learning about the misalignment angle,
and reducing the CEP. The most significant result is the ay value
which decreases from 15.62 for a = 0. to 2.78 for a = 0.1. This in
turn decreases the CEP significantly.
The cost is rising steadily but this is due to the integral term of
control. It is hypothosized that the weighting value for the control
integral is too high. The significant result is that a small amount of
37
maneuvering will minimize the CEP.
Figures 12 through 15 show graphically the results of Case 1.
Figure 12 is a schematic of four flight paths for a = 0.1, a = 0.3,
a = 0.6, and ot =1.0. The general trend of the results are evident
and point to the conclusion that a small turn minimizes the CEP. The
time of flight is a significant factor. For a = 0, TOF = 100., for
a = 1.0, TOF = 115.68. During Phase II, the covariances are increasing
and every second of flight is significant. Thus, the optimal tradeoff
between maneuvering versus TOF, as shown by Case 1 is a small turn that
does not increase the TOF significantly. The TOF for ot = 0.1 is 100.222.
Table 1. Result of Case 1 Analysis
a
TOF
ax
ay
CEP
COST
P18
0
100.0
8.20
15.62
14.00
14.00
1.0E-4
0.1
100.2
10.91
2.78
8.05
58.11
2.69E-6
0.2
100.9
14.52
3.70
10.71
61.85
5.39E-8
0.3
102.2
15.02
4.64
11.56
65.17
5.83E-7
0.4
102.3
13.89
4.98
11.09
64.81
3.07E-7
0.5
104.7
15.38
6.05
12.61
66.63
2.56E-7
0.6
106.1
13.62
6.37
11.75
66.75
1.80E-7
0.7
109.1
15.40
7.30
13.35
69.86
1.31E-7
0.8
109.9
13.13
7.32
12.02
71.30
9.9EE-8
0.9
112.9
14.73
8.11
13.43
72.20
8.74E-8
1.0
115.6
14.40
8.50
13.47
73.75
7.07E-8
0.20
0.40 0.60
0.80
r.oo
ALPHA
-
Figure 13. a% vs a for Case 1
40
,0.00 2.50 5.00 7.50
.00 0.20 0.40 0.60 0.80 1.00
, ALPHA
Figure 14. <7y va a for Case 1
41
Case 2
Radar Range = 40,000 m.
Xf = 201,000. m.
XD = 1000. m.
The major results for Case 2 are presented in Table 2. With no
maneuvering, a = 0. the terminal CEP is 38.80. This decreases only
slightly to 38.16 with a =0.1. The general trend of Case 1 is again
evident in Case 2 but to a lesser degree. The minimum CEP is again
attained by a small turn and small amount of turning. The ay value again
shows the sharpest decrease going from 37.72 for a = 0. to 14.12 for
a = 0.1. Figures 16 through 19 show graphically the results of Case 2.
The CEP does not decrease as favorably as in Case 1. This may be
due to an increased TOF and less time during Phase I for learning about
the misalignment angle. As seen from Table 3, the values of ax and Oy
are greater in Case 2 than in Case 1 at the end of Phase I flight. The
covariance of the misalignment angle is also greater in Case 2. This
coupled with a longer flight time in Phase II in Case 2 diminishes the
dual control nature of the problem.
Table 2. Results of Case 2 Analysis
a
TOF
ax
ay
CEP
COST
00
0
200.
28.26
37.72
38.80
38.80
l.OE-4
0.1
200.2
50.76
14.12
38.16
88.34
2.1E-6
0.2
200.8
95.09
22.36
69.06
121.8
4.9E-6
0.3
201.8
98.15
26.59
73.34
125.3
3.2E-6
0.4
202.4
93.99
28.57
72.06
125.5
2.1E-6
0.5
203.9
87.98
32.61
70.91
125.2
1.2E-6
0.6
205.2
96.99
40.73
90.98
137.67
1.1E-6
0.7
207.7
111.82
50.97
95.72
153.03
1.0E-6
0.8
208.5
84.21
42.98
74.78
125.19
5.6E-7
0.9
1.0
211.0
102.21
53.61
91.62
151.85
6.3E-7
213.5
97.57
56.33
90.49
151.87
5.0E-7
43
Figure 16. Flight Paths for Case
°0.03
0.20
0.40 0.60
0.60
1.00
ALPHA
Figure 18. vs a for Case 2
46
-20.00 32.50 45.00 57.50
u i i - • l
Table 3. Comparison of Case 1 and Case 2 at End of Phase I
Case 1
i
Case 2
’ a
0
.1
.5
1.0
0
.1
.5
1.0
j ax
11.8
11.8
13.8
12.3
10.2
15.1
24.6
21.4
ay
54.0
0.15
3.0
6.9
44.0
0.2
4.9
12.0
P18
1.0E-4
2.1E-6
2.7E-7
7 . 4E-8
1.0E-4
2.9E-6
1.4E-6
5.4E-7
These two examples show that dual control can be applied to an air-
to-ground missile with a constant misalignment angle. These examples
also show significant results for all maneuverine during Phase I. Cases
were evaluated for accomplishing the final turn after Phase I. These
results are far less encouraging, however, they do show a significant
result: that high -g maneuvering during a non-radar environment can
increase the terminal CEP. Table 4 shows the results for Case with the
final turn initiated during Phase II rather than in Phase I.
Table 4. Results of Case 1, Terminal Maneuvering During Phase II
a
TOF
X
y
CEP
COST
P18
0
100.0
8.19
15.62
14.00
14.00
1.0E-4
0.1
100.9
113.06
6.78
70.46
80.06
9.8E-5
0.2
103.2
215.23
16.35
136.17
148.25
9.1E-5
0.3
106.6
288.28
20.17
181.37
193.61
6.5E-5
0.4
109.6
289.82
22.38
183.57
199.34
4.3E-5
0.5
115.3
243.82
22.26
156.46
175.80
1.8E-5
0.6
118.1
211.13
27.79
140.49
159.46
1.0E-5
0.7
122.8
173.60
32.40
121.13
142.36
4.8E-6
0.8
128.9
149.15
30.25
105.49
130.63
2.7E-6
0.9
136.7
128.20
28.41
92.38
122.38
1.6E-6
1.0
143.9
109.53
30.20
82.17
116.32
9.0E-7
As seen from the table, a small turn will not reduce the terminal
CEP when the final turn is completed outside the radar environment.
48
*---
However, the trend of the data indicates that a large initial turn will
offset the maneuvering during Phase II. Note that the CEP reaches a
maximum at a = 0.4 and then decreases steadily as a increases to 1.0
radian. This is also related to the covariance of the misalignment angle.
When the P^g state is sufficiently small then the CEP is reduced.
The results of this chapter have demonstrated the feasibility of
dual control as applied to an air-to-ground missile. However, the method
of maneuvering was severely restricted. Chapter IV investigates the
problem using a first order gradient technique which allows more flexi-
bility during Phase I.
IV. Gradient Optimization Technique
Theory
To understand the numerical first-order gradient technique, it is
first necessary to develop an understanding of the optimization problem,
the general gradient methods, and special variations due to the parti-
cular class of problem being considered.
The general problem is to find the control history of the m-dimen-
sional control vector u(t) which will bring the performance function J
to a minimum. The control function is expressed as:
J = 0[x(tf),tf] + j' L[x,u,t]dt
The system equations are represented as:
(65)
x = f(x,u,t) (66)
where x is an n-vector.
The general method consists of adding Lagrange multipliers to the
cost function. This becomes:
J = 0[x(tf),tf] +/[L + At(f - x) ]dt
Defining the Hamiltonian H as:
H = L + ACf
the cost function becomes:
J = 0[x(tf),tf] +/(H- Atx)dt
= 0 + /»dt - / \Cxdt
Integrating the last term of Equation 69 by parts yields:
y^xdt = At(tf)x(tf) - At(t0)x(t0) ~ t)x(t)dt
(67)
(68)
(69)
(70)
50
thus:
L
/t £
(H(x,u,t) + Xt(t)x(t) }dt (71)
Considering a variation in J due to variations in the control vector
u(t) for fixed tQ and t^:
6J = [(H Xt)sx] + [xtsx]t0
/tf[(S-c)
(72)
9h
Sx + Su(t) dt
A necessary condition for a minimum to exist is that <5J = 0 for arbitrary
<Ju(t). Therefore, from Equation 72 a necessary condition is that
3H
X3X
(73)
with
X(tf) - (tf)
(74)
and
3H = 3L ,t _3f
3u 3u 3u
(75)
In summary, a two-point boundary value problem must be solved to find
the minimizing value of J. The differential equations that must be
solved are:
x = f(x,u,t) (76)
»•- - - - <%*>* - <‘X>
3fy
,3L/ vt
(77)
where u(t) is determined from
X " 0
(78)
51
Considering no variations in the initial conditions. Equation 72 becomes:
ij .y|a judt .
To apply this gradient technique to the problem of this study, it is
necessary to consider the terminal constraint ip[x(tf)]. This constraint
may be written as follows:
pi, ♦ X)
dt
(79)
*[x(tf)] =
xl(tf)
x (tf)
(80)
In a similar manner to the previous development, Lagrange multipliers
are added to influence the terminal constraint. This may be developed
using a special cost function of the form = <J>[x(tf)]. Designating
the coefficient as R, letting ^(x(t^)) = 0(x(t^)), and considering <5x(t0) =
0, Equation 72 becomes:
i3i - <% - + /[<!! + + 15
choosing R to make the coefficients of <JX zero yields
dt
(81)
Rt = _ 3H/ _ 31 3L
R /3x ^x R ' /3x
3f.
3L
(82)
However, this coefficient is considering a special performance index, J^,
where L = 0. Therefore,
' *' 5 ■ - X *
(83)
with
*'<*£> - /&<*£>
(84)
Considering the special cost function J^, Equation 79 becomes:
52
(85)
<5 u(t)dt
Now a control history may be formulated that decreased J and satisfies
the terminal constraint. Multiplying Equation 85 by a constant V and
adding to Equation 79 yields:
<5 j + v1,5xi(tf) = y f {U + l> + viR]t /{J <5u(t)dt (86)
*■0
Now choose
«“(*> - -K jf^u)'[X + uiRl + (1^)1 (87>
This forces the ^x(t^) to vanish. The v's may be chosen to
satisfy the terminal constraint.
Application to Problem
In applying the theory of gradient optimization to the problem of
minimizing the terminal CEP of an air-to-ground missile with constant
misalignment angle, numerical difficulties are encountered. The first
attempt was to use an unspecified final time and allow gradient search
during Phase I while commanding the missile to fly toward the target
during Phase II. This method had difficulties in the backward integration
of the influence functions. Therefore, to make the gradient technique
more applicable, the problem was reformulated slightly.
The major change is to project the cost at the end of Phase II based
on the system state at the end of Phase I. This can be accomplished since
the P equation is linear during Phase II. Thus,
P(tf) »4>(t,tg) P4>(t,tg) + <t>(s)Q <J)(s)ds (88)
53
I
where
t = time at end of Phase I
r
t = t. - t
g f r
If the final turn to the target has been accomplished during Phase I,
then there will be zero nominal lift during Phase II and the system
equations may be expressed as:
0 0 10 0
0 0 0 1 0
Z 0 0 C 0
0 0 0 0 0
0 0 0 0 0
Thus, during Phase II:
<l>(t,tg) = e
1 0 t 0 0
0 1 0 t 0
0 0 10 0
0 0 0 1 0
0 0 0 0 1
Substituting Equation 90 into Equation 88 yields:
P(tf)
P11 + 2tg + Cg P33
P12 + tgP25 + tgP14 + tg P34
P,9 + t„Pi/ + + t2 P34 P_„ + 2t P,, + t2Py/
12 g 14 g 23 g 22 g 24 g 44
54
dt
+
t2.io4
0
The cost at the terminal time may now be written as a function of the
state at the end of Phase I. This becomes
J(t£) - 0.588 [Vpu + 2tgPu + tg2 P33 + tg3 \ ' 10 4
+VP22 + 2tgP24 + 2tg P44 + ' 10~4
(92)
t„
/ M":
dt
By choosing ip[x(tr)] to satisfy the constraint of the terminal guidance
used in Chapter III, all maneuvering will be accomplished during Phase I.
Thus,
H*[*(tr)] = [0req - X3] (93)
To make the problem more realistic for an air-to-ground scenario, the
final target position was increased from 101,000. m. , as used in Chap-
ter III to 601,000. m. The maneuvering time for this method was chosen
at a constant 60.0 seconds. Thus, Phase I flight will continue for 60.
seconds then the projected cost will be evaluated.
Bryson and Ho (1:222) outline a method for a first-order gradient
technique for problems with some state variables specified at a fixed
terminal time. By specifying the terminal constraint as a function of
the Q or state this method is applicable to the problem. The general
method is as follows:
55
I
1. Estimate a set of control histories, u(t).
2. Integrate the system equations forward with the specified initial
conditions x(tQ) and estimate from Step 1. Record x(t), u(t),
^[x(tf)], J.
3. Determine the n-vector of influence functions, c(t), and the
nxq matrix of influence functions, R(t), by backward integration
of the influence functions.
.3f . ,t ,9L / .t
c = -( Ax> C " ( At)
- <8>3x1)e£
* - - < /k>R
4. Compute the following integrals.
5. Choose values of Sty to cause the next nominal solution to be
closer to the specified ^[x(tf)].
d'P = “ Ei|i[x(tf ) ]
” - - Ciw]_1 M* + V
6. Repeat Steps 1 through 5 using an improved estimate of u(t)
where
56
(1 i = j
R(tf) = )0 i f j
I
<Ju(t) = [w(t)] 1 3_^ + (p(t) + r(t)v)t |^J
stop when
4>[x(t . ) ] = 0 and I T T - I T . I , ,_1I , T = 0
rL j J JJ Jip <Jnp ipJ
This algorithm was applied to the thesis problem. Appendix A con-
tains the details of the influence functions. Step 3, and the 18 x 18
9f
matrix, /9x. Figure 20 presents a flow chart of the first-order gra-
dient method. The actual computer listing is contained in Appendix B.
Three cases were evaluated using this method. The cases differ
in the initial guess of the control history for u(t). The three cases
evaluated were:
Case 1
This case was chosen because of its simplicity and minimum trajectory
to initialize the gradient algorithm.
Case 2
57
Figure 20. Flow Chart for First Order Gradient Technique
58
I
This case represents the minimum from Chapter III. Perhaps this
trajectory can be improved by the gradient algorithm.
Case 3
'
(
It was felt that perhaps a series of maneuvers would be the ideal
trajectory for a global minimum. This initial guess should be able to
demonstrate the effects of a constant turning trajectory.
The following values were computed and reported for results:
TOF = Time of flight, this includes the predicted time of flight
during Phase II.
a = Covariance of x-estimate. For this method, it is recorded
as the value at the end of Phase I.
a
y
= Covariance of y-estimate. For this method, it is recorded
as the value at the end of Phase I.
CEP = Circular Error Probable. The area where 50% of the trajec-
tories should terminate. As explained earlier, this is the
projected value at the end of Phase II.
Cost = Total performance function. This is also the projected cost
at the end of Phase II.
CTy = Covariance of misalignment angle. For this method, it is
recorded as the value at the end of Phase I.
PSI = The terminal constraint at the end of Phase I. W * (0
„ x re9
Case 1
The major results of Case 1 are summarized in Table 5. The results
are interesting in that the initial reduction of CEP by almost 100 m.
59
is achieved by turning the missile at a negative 2g turn at about 10
seconds into Phase I. Only slight changes are made in this flight path
after the initial gradient search. After ten cycles, the CEP is reduced
to 121.65 m. This is a substantial reduction from the initial CEP of
227.23 ra. The method diverges quickly for the ip[x(tg)] but slowly is
trying to bring the \p value to a minimum. In cycle 10, the value is
-.1323 rad. or about 7° off a straight- in shot to the final target.
Case 2
The major results of Case 2 are summarized in Table 6. The results
do not show such a significant change in control history as in Case 1
but there is still a substantial amount of maneuvering at approximately
t = 10. seconds. The algorithm produces almost no changes after t =
25 seconds.
After eight cycles, the CEP has been reduced to approximately 130
m. The major changes are to reduce y. This verifies the results of
Chapter III in that a small initial maneuver substantially reduces the
CEP. As seen from Table 6, the CEP has only been reduced by 3 m. while
in Case 1 the CEP was reduced by 100 m.
Case 3
The major results of Case 3 are summarized in Table 7. The results
show a substantial decrease in CEP from an initial 308.44 to 123.77 in
Cycle 7. This is mostly from a modification of the input value.
An interesting result of this case is the large jump seen in Cycle
8. The algorithm seems to be searching for a method to reduce Oy . The
algorithm finally commands an opposite lift vector at t = 10. seconds.
This increases the CEP slightly. The method the • ran into numerical
60
integration problems and stopped.
600.40 3.02 8.18 121.47 121.65 .0048 -.1323
Figure 22. CEP vs Cycle-No. for Case 1
64
CEP COST 0., PSI
CO
ON
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00
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601.14 3.76 8.04 130.22 132.22 1.327E-3 -.2548
CYCLE-NO
Figure 25. CEP vs Cycle-No. for Case2
68
'll 00
2.00
3.00 4.00
5! 00
6 00
CYCLE-NO
Figure 26. PSI vs Cycle-No. for Case 2
69
70
00 ‘002 00*081 00*091 00*07*
Figure 28. CEP vs Cycle-No. for Case 3
72
r. Conclusions and Recommendations
The theory of dual control has been applied to an air-to-ground
missile with a constant misalignment angle. The analysis, although
modeled simply, has shown significant results. The data from Chapter
III and Chapter IV show the inherent correlation between the knowledge
of the error parameter and the terminal CEP of the missile.
There are two major conclusions that can be drawn.
1. Maneuvering during a radar environment decreases the covariance
of the unknown misalignment parameter.
2. Because of this learning about the error parameter, the CEP of
the missile can be substantially reduced.
From Chapter III, it was also learned that high-g maneuvers outside
the radar environment can amplify the navigation errors greatly. It is
therefore recommended that any air launched ballistic missile perform
its required maneuvering, especially high-g turns, while it is still in
the radar environment of the carrier aircraft. The tracking information
from the radar system will aid greatly in reducing the terminal state of
INS systems.
From Chapter IV, it is evident that there may be many local minimums.
Hence, it may not be concluded that this research has discovered an abso-
lute best trajectory for a minimum CEP. Many more cases of input control
guesses can be made which may further reduce the CEP. However, the basic
concept of dual control has been successfully applied.
Further research in this area is needed for a more comprehensive and
complete understanding of this theory. The following recommendations are
made for future study:
1. Radar measurements should be modeled as discrete measurements.
74
2. Three dimensional analysis would allow greater maneuverability
and more error parameters to be identified.
3. Initial boost phase could be modeled. This is an area of
increased observability.
The theory of dual control may be able to make significant contri-
butions to inertial navigation systems. It will certainly aid in getting
the best performance that each system is capable of producing.
j
I
l
Li
75
Bibliography
1. Bryson, A. E., and Ho, Yu-Chi. Applied Optimal Control, Washington,
D.C.: Hemisphere Publishing Corporation, 1975.
2 . Dayan , R . Application of a Maximum Likelihood Parameter Estimator
to an Advanced Missile Guidance and Control System, Thesis, Wright -
Patterson Air Force Base, Ohio: Air Force Institute of Technology,
December, 1977.
3. Eykoff, P. System Identification. London: John Wiley and Sons,
1974.
4. Feldbaum, A. Optimal Control Systems. New York: Academic Press,
1965.
5. Hornbeck, R. W. Numerical Methods, New York, N.Y.: Quantum Pub-
lishers, Inc., 1975.
6. Lipshutz, S. Probability, New York: McGraw-Hill Book Company,
1965.
7. Maybeck, P. S. Stochastic Estimation and Control, Part I, Unpub-
lished Notes. Wright-Patterson Air Force Base, Ohio: Air Force
Institute of Technology, February, 1975.
8. Reid, J. G. Sensitivity Operators and Associated System Concepts
for Linear Dynamic Systems, Technical Report AFAL-TR-76-118 .
Wright-Patterson Air Force Base, Ohio: Air Force Institute of
Technology, July, 1976.
9. Tse, E. and Y. Bar-Shalom. "Wide-Sense Adaptive Dual Control for
Nonlinear Stochastic Systems," IEEE Trans. Auto. Control, AC-18:
98-108, April, 1973.
76
Appendix A
/
This appendix presents the 18 x 18 matrix, . This is shown on
pages 81 through 89. Also influence function equations are presented.
These are described in Chapter IV as:
and
where
c is a n-vector
R is a nxq matrix
77
C3 = C1-V-Sin(X3) - C2-V.Cos (X3)
C4 = C4* (2AX4 + 2BX5) + C5(AX5 + BXg) + C6(AX6 + BX1Q) + C?
(AX? + BX1X) + Cg (AXg + BX12)
C5 = C4(2BX4 + 2DX5) + C5(AX4 + 2BX5 + DXg) + CgCBXg + DX1Q)
+ C7(BX? + DX1X) + Cg (BXg + DX12) + C9(2AX5 + 2BXg) + C1Q
(AXg + BX1q) + C11(AX? + BXn) + C12(AXg + BX12)
C6
= -2C4 + Cg(AX4 + BX5)
+ C1q(AX,
i + BV
+ C13(2AX6
+ 2BX10)
+ C14(AX? + BX1X) +
C15(AX8 +
“l2>
r
K
= -c5 + c?(ax4 + bx5)
+ Cn(AX5
+ bx9) + c14(ax6 +
BX10>
+ C16(2AX? + 2BXX1)
+ C17(AX8
+ “l2>
'C8
" L2C6 " L1C7 + C8(AX4
+ BX5) +
c12<axs
+ BXg) + C.
L5<M6
+ BX1(J) + C17(AX? + BXn) + C18(2AXg + 2BX12)
C9 = C5(BX4 + DX5) + C9(2BX5 + 2DX9) + C^BXg + DX1Q)
+ CU(BX? + DXL1) + C12(BXg + DX12)
C10 " " c5 + C6(BX4 + DX5) + C10(BX5 + DV + C13(2BX6 + 2DX10)
+ C14(BX? + DX1L) + C15(BXg + DX12)
CX1 - C?(BX4 + DX5) - 2C9 + CX1 (BX5 + DX9) + C14(BX6 + DX10)
78
I
+ C16(2BX? + 2DXn) + C17(BXg + DX12)
C12 = C8(BX4 + DX5) + L2C10 ~ L1C11 + C12(BX5 + DV + C15(BX6
+ DX1q) + C17(BX? + DX1X) + Clg(2BXg + 2DX12>
C13
~
'C6
c
=
-C -
c
14
7
10
•
__
c_ +
2L„C
C15
8
2 13
C ,
- c
16
11
C „
- r
+ L C ,
17
12
2^14
C „
_
L„C ,
- L C
18
2 15
1 17
I
R3 = - v Sin(X3)
all remaining R's will equal zero because of initial conditions.
-2Ax, -2Bx
o
-Bx , -Dx
A0-A058 515 AIR FORCE INST OF TECH MRISHT-PATTERSON AFfi OHIO SCH— ETC
DUAL CONTROL ANALYSIS OF AN AIR TO OROUNO MISSILE* (U)
MAR 75 J P KAUPPILA
UNCLASSIFIED AFIT/SA/EE/7S-1
F/§ 16/4*1
>-
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116
VITA
James P. Kauppila was born in Hancock, Michigan on June 15, 1948.
He was raised in Jackson, Michigan. After graduating from Jackson High
School, he was appointed to the United States Air Force Academy. After
graduation from the Academy in 1970, he went to pilot training at
Williams Air Force Base, Arizona. He then became a B-52H copilot and
later upgraded to Aircraft Commander. He came to AFIT in June of 1976.
SECURITY CLASSIFICATION of This RACE fH7..n Dalo Entered)
REPORT DOCUMENTATION PAGE
READ IN: . RUCTIONS
BEFORE COMI-I.E'; I.NG FORM
t. REPORT NUMBER
AFTT/OA/BS/78-1
/
2. GOVT ACCESSION NO
3. RECIPIENT’S CATALOG NUMBER
4 TITLE (end Subtitle)
DUAL CONTROL ANALYSIS OF AN
AIR TO GROUND MISSILE
5. TYPE of REPORT ft PERIOD COVERED
MS Thesis
6. PERFORMING ORG. REPORT NUMBER
7. AUTHORfs)
James P. Kauppila
Ca^t USAF
8. CONTRACT OR GRANT NUMBERS)
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Air Force Institute of Technology (AFIT-EN)
Wright-Patterson AFB, Ohio 45433
10. PROGRAM ELEMENT, PROJECT, TASK
AREA a WORK UNIT NUMBERS
11. CONTROLLING OFFICE NAME AND ADDRESS
12. REPORT DATE
March 1 97S
13. NUMBER OF PAGES
117
14. MONITORING AGENCY NAME ft ADDRESSf// different from Controlling Ol/ice)
15. SECURITY CLASS, (of this report)
Unclassified
I5a. DECLASSIFICATION DOWNGRADING
SCHEDULE
16. DISTRIBUTION STATEMENT (ot this Report)
Approved for public release; distribution unlimited
J *7. DISTRIBUTION STATEMENT (ol the abstract entered In Block 20, It different from Report)
18. SUPPLEMENTARY NOTSS t 4
/{pta-oved release: I AW AFR 199-17
JERjWfiL pf GUESS,-' Capi?ain7 USAF
Director of Information
\
19. KEY WORDS (Continue on reverse side II necessary end Identity by block number;
Dual Control
Optimal Control
Parameter Identification
Sensitivity
TcT ABSTRACT (Continue on reverie ride II necoir.O' end I dr nitty by block number)
Many errors are known to exist in Inertial Navigation Systems of
modern air-to-ground missiles. These error sources, if undetected, con-
tribute to navigation errors of position and velocity. This study
analyses one source of INS errors — the misalignment of the accelero-
meter reference frame. By maneuvering a missile, the error source
becomes more observable. Thus, a better estimate can be made of the
error source. This directly influences the estimate of position. — Cor ^
dd
FORM
AN 73
1473
EDITION OF I NOV 6S IS OBSOLETE
-DHGI.ASSIFISIL
SECURITY CLASSIFICATION OF THIS PAGE fRFpn Dele Entered)
FIT'D
SEClIPITY CLASSIFICATION CF THIS P AGECOTion Data Entered)
Hence, in order to minimize the terminal navigation error, some control
energy must be expended to identify the error source. This dual control
problem may be viewed as an optimization problem. By formulating a
performance index of the terminal error and control energy appropriate
mathematical techniques should yield an optimal flight trajectory.
This thesis seeks to analyze the dual control nature of an air-to-
ground missile. Tv?o methods are used. The first uses a predetermined
flight rath which is incremented until a minimum is reached. The second
is a first-order gradient vhich allows greater freedom in the control
law.
SECURITY CLASSIFICATION OF THIS PAGEdFh.n Dal* Entered)
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